Properties

Label 4-1050e2-1.1-c1e2-0-6
Degree $4$
Conductor $1102500$
Sign $1$
Analytic cond. $70.2963$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 6-s + 5·7-s + 8-s − 6·11-s − 8·13-s − 5·14-s − 16-s + 3·17-s + 4·19-s − 5·21-s + 6·22-s − 3·23-s − 24-s + 8·26-s + 27-s − 12·29-s − 5·31-s + 6·33-s − 3·34-s − 8·37-s − 4·38-s + 8·39-s − 6·41-s + 5·42-s + 16·43-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.408·6-s + 1.88·7-s + 0.353·8-s − 1.80·11-s − 2.21·13-s − 1.33·14-s − 1/4·16-s + 0.727·17-s + 0.917·19-s − 1.09·21-s + 1.27·22-s − 0.625·23-s − 0.204·24-s + 1.56·26-s + 0.192·27-s − 2.22·29-s − 0.898·31-s + 1.04·33-s − 0.514·34-s − 1.31·37-s − 0.648·38-s + 1.28·39-s − 0.937·41-s + 0.771·42-s + 2.43·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1102500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(70.2963\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1050} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1102500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3248839849\)
\(L(\frac12)\) \(\approx\) \(0.3248839849\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.21594968828720264315636680607, −9.591034366353285692457886675475, −9.520080058093399295269963142106, −8.771558479222400954870612232997, −8.534978071025457659767133259231, −7.76592485177940369073543868461, −7.57710935987023284350534813715, −7.43013666605564145928931680897, −7.33678282902933142419076155438, −6.18059737922566810561186631509, −5.54121231432105682420240270366, −5.45142226532125361248720282747, −4.96093199284181388781088370669, −4.72210932703943031001834966506, −4.13079696271995107313154256775, −3.20123659091384495349033967665, −2.69936387284335889343650989882, −1.77567645663053153154022325902, −1.74477772940290172180282620313, −0.28967426592619811781094117761, 0.28967426592619811781094117761, 1.74477772940290172180282620313, 1.77567645663053153154022325902, 2.69936387284335889343650989882, 3.20123659091384495349033967665, 4.13079696271995107313154256775, 4.72210932703943031001834966506, 4.96093199284181388781088370669, 5.45142226532125361248720282747, 5.54121231432105682420240270366, 6.18059737922566810561186631509, 7.33678282902933142419076155438, 7.43013666605564145928931680897, 7.57710935987023284350534813715, 7.76592485177940369073543868461, 8.534978071025457659767133259231, 8.771558479222400954870612232997, 9.520080058093399295269963142106, 9.591034366353285692457886675475, 10.21594968828720264315636680607

Graph of the $Z$-function along the critical line